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7.1 Definitions and notations Reactive System The intuition is that a transition system consists of a set of possible states for the system and a set of transitions - or state changes - which the system can effect. When a state change is the result of an external event or of an action made by the system, then that transition is labeled with that event or action. Particular states or transitions in a transition system can be distinguished.

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model to describe the behavior of systems digraphs where nodes represent states, and edges model transitions state:  the current color of a traffic light  the current values of all program variables + the program counter  the value of register and output transition: (“state change”)  a switch from one color to another  the execution of a program statement  the change of the registers and output bits for a new input

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These two systems are weak isomorphic Theorem:  If two transition systems are isomorphic, then they are weakly isomorphic.  Weak isomorphism is an equivalence relation

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Example two isomorphic TS are bisimilar, but bisimilar TS are not necessarily isomorphic

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The lady or the tiger

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Strong Isomorphism: the transition systems are identical except for the names of the states. Weak Isomorphism: the transition systems are strongly isomorphic provided that the transition systems are restricted to the reachable states. Bisimulation Equivalence: the transition systems have the same behavior, and make choice at same time.

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Use TS to present the behavior of all the modeling language Then Use TS to prove the equivalence respectively

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The free product of transition systems

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p q s t p,s p,t q,s q,t

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The free product assumes that in a global system, all component systems execute their transitions simultaneously, it is possible to divide time into intervals in such a way that during each of those intervals each component executes exactly one transition. In other words, the same ‘clock’ drives the different transition systems forming the product.

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The synchronous product of transition systems When processes interact, not all possible global actions are useful, since the interaction is subject to communication and synchronization constraints. The transition system associated with the system of processes must therefore be a subsystem of the free product of the component transition systems. The communication and synchronization constraints that define the subsystem can always be simply expressed by the synchronous product, formally defined below.

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p q s t p,s p,t q,k q,t k p,k q,s

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a b c p q s t p,s p,t q,k q,t k p,k q,s

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a b c p q s t p,s p,t q,k q,t k p,k q,s

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p q s t p,s p,t q,s q,t

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Shared label

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Modeling sequential circuits

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A Mutual Exclusion Protocol Two concurrently executing processes are trying to enter a critical section without violating mutual exclusion

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State Space The state space of a program can be captured by the valuations of the variables and the program counters For our example, we have  two program counters: pc1, pc2, domains of the program counters: {out, wait, cs}  three boolean variables: turn, a, b, boolean domain: {True, False} Each state of the program is a valuation of all the variables

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Each state can be written as a tuple (pc1,pc2,turn,a,b) Initial states: {(o,o,F,F,F), (o,o,F,F,T), (o,o,F,T,F), (o,o,F,T,T), (o,o,T,F,F), (o,o,T,F,T), (o,o,T,T,F), (o,o,T,T,T)} – initially: pc1=o and pc2=o How many states total? 3 * 3 * 2 * 2 * 2 = 72 exponential in the number of variables and the number of concurrent components

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Transition Relation specifies the next-state relation, i.e., given a state what are the states that can come immediately after that state For example, given the initial state (o,o,F,F,F) Process 1 can execute: out: a := true; turn := true; or Process 2 can execute: out: b := true; turn := false; If process 1 executes, the next state is (w,o,T,T,F) If process 2 executes, the next state is (o,w,F,F,T) So the state pairs ((o,o,F,F,F),(w,o,T,T,F)) and ((o,o,F,F,F),(o,w,F,F,T)) are included in the transition relation

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Temporal Properties  once r is 1, it will be 1 forever  Two program cannot in the critical section together  If you choose sprite, you cannot get beer unless you pay again  No deadlock

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Introduction Temporal logic is a formalism for describing sequences of transitions between states in a reactive system. Properties like eventually or never are specified using special temporal operators. CTL*  Software Engineering Group  53

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X (next time) requires the property holds in the second state of the path F (eventually) the property will hold at some state on the path G (always) the property holds at every state on the path U (until) if there is a state on the path where the second property holds, at every preceding state, the first one holds R (release) the second property holds along the path up to and including the first state where the first property holds. However, the first property is not required to hold eventually

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two types of formulas in CTL*  state formulas ( which are true in a special state )  path formulas ( which are true along a special path ) syntax of state formulas rules:  if then p is sf  if f and g are sf, are sf  if f is a pf, then E f and A f are sf  Software Engineering Group  58

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syntax of path formulas:  if f is a sf, then f is also a pf  if f and g are pf,, X f, F f, G f, f U g and f R g are pf CTL* is the set of state formulas generated by the above rules semantics of CTL*  if f is a sf, M, s ->f means that f holds at state s in the M  if g is a pf, M, π-> g means that g holds along path π in the M  Software Engineering Group  59

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CTL and LTL two sublogics of CTL*  branching-time logic the temporal operators quantify over the paths that are possible from a given state. Temporal operators must be immediately preceded by a path quantifier. if f and g are sf, X f, F f, G f, f U g and f R g are pf A(FG p)  Linear temporal logic operators are provided for describing events along a single computation path. LTL implicitly quantifies universally over paths. If, then p is pf, Af where f is a pf AG(EF p)  Software Engineering Group  62

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Examples Let "P" mean "I like chocolate" and Q mean "It's warm outside."  AG.P "I will like chocolate from now on, no matter what happens.“  EF.P "It's possible I may like chocolate some day, at least for one day."  AF.EG.P "It's always possible (AF) that I will suddenly start liking chocolate for the rest of time." (Note: not just the rest of my life, since my life is finite, while G is infinite).  EG.AF.P "This is a critical time in my life. Depending on what happens next (E), it's possible that for the rest of time (G), there will always be some time in the future (AF) when I will like chocolate. However, if the wrong thing happens next, then all bets are off and there's no guarantee about whether I'll ever like chocolate."  Software Engineering Group  66

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A(PUQ) "From now until it's warm outside, I will like chocolate every single day. Once it's warm outside, all bets are off as to whether I'll like chocolate anymore. Oh, and it's guaranteed to be warm outside eventually, even if only for a single day." E((EX.P)U(AG.Q)) "It's possible that: there will eventually come a time when it will be warm forever (AG.Q) and that before that time there will always be some way to get me to like chocolate the next day (EX.P)."  Software Engineering Group  67